Systems and Methods of Network Analysis and Characterization

ABSTRACT

Systems and methods for characterizing networks are disclosed. In several embodiments, a network analyzer applies a network analysis to a network that replaces components of the network in a model of the network with equivalent or bounding models. The network analyzer can then characterize the simplified model of the network and an assessment can be made concerning the accuracy of the characterization of the network.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a continuation of U.S. patent application Ser. No.13/364,195 entitled “Systems and Methods of Network Analysis andCharacterization” to Effros et al., filed Feb. 1, 2012, whichapplication claims priority under 35 U.S.C. §119(e) to U.S. ProvisionalPatent Application Ser. No. 61/481,648, filed May 2, 2011, thedisclosure of which is incorporated by reference herein in its entirety.

STATEMENT OF FEDERALLY SPONSORED RESEARCH

This invention was made with government support under W911NF-07-1-0029awarded by DARPA, CCF-1018741 awarded by the National ScienceFoundation, and FA9550-10-1-0166 awarded by Air Force (AFOSR). Thegovernment has certain rights in the claimed inventions.

FIELD OF THE INVENTION

The present invention generally relates to network analysis and morespecifically relates to the calculation of inner and outer bounds oncritical network characteristics such as capacity and jointsource-channel coding regions and to the creation of simplified networkmodels that preserve or bound such properties from the original network.

BACKGROUND

A communication network includes a collection of communicating devices(e.g., telephones, moderns, computers, PDAs, video game consoles,routers, sensors) that can transmit and receive signals through avariety of communication media (e.g., wires, cables, air, water, space)for the purpose of delivering data (e.g., text, images, audio, video,sensor measurements) across space or time. There are a wide variety ofcommunication networks in use today. These include, but are not limitedto, traditional (wireline) telephone networks, wireless phone networks(e.g., cellular phone networks, satellite phone networks), local areanetworks (LANs), wide area networks (WANs), the Internet, cabletelevision networks, RFID networks, and sensor networks.

There are many metrics used in characterizing the performance of anetwork. One common measure is the maximal data rate at which thenetwork can deliver information. (Maximal data rates are sometimesdescribed using terms such as the “network bandwidth,” “databand-width,” “information rate,” or “network capacity.” The term“bandwidth” in the context of data networks is typically used toquantify the amount of data delivered per unit of time. The term“capacity” is used to refer to the maximum possible data rate orcollection of data rates that a link, path, or network can deliver perunit time.) In other cases, it may be useful to describe the minimaldistortions with which a network can deliver a collection of sources totheir intended destinations, the latency or delay suffered ininformation delivery, the robustness of the network to componentfailures or adversarial attack, the extent to which information can beprotected against eavesdropping, and so on. The metric of interest inanalyzing a given network is often highly dependent on the application.For example, transmissions of live voice and video are far moresensitive to delay than is email, even when both data types travelthrough the same network.

Analyzing a network's performance is useful for a variety ofapplications. These include but are not limited to the followingexamples. Network users who wish to compare a variety of availablenetworks can use network analysis to determine which network will givethe best performance for their needs. Network designers interested inbuilding new networks or reconfiguring old ones can use network analysisto optimize design choices with respect to relevant network metrics.Network analysis is also of potential use for guiding research anddevelopment. For example, by analyzing both metrics that providetheoretical bounds on a network's best possible performance and metricsthat measure current performance using existing algorithms, researchersand entrepreneurs can distinguish between problems for which currentsolutions already achieve good performance and those where large gapsbetween theory and practice suggest a greater potential for advance.Network service providers can use network analysis to discoverunderlying structure in their networks, understand how much enabling anew service impacts existing services, inform pricing, guide networkdesign, and improve network upgrades. Techniques for estimating networkperformance are also important for overlay network routing, trafficengineering, and capacity planning support.

Theoretical analysis of network performance is often extremelydifficult. For example, the capacity of a single memoryless, noisypoint-to-point channel is well-characterized, but the capacities formany memoryless three-node networks (e.g., non-degraded broadcast andrelay channels), most networks that can be built from memoryless, noisypoint-to-point channels, and most channels with memory remain unsolved.A key challenge is that the capacities of a network's components do notnecessarily determine the capacities of networks that can be built fromthose components. For example, consider a 4-node network in which node 1transmits to nodes 2 and 3 through a broadcast channel and nodes 2 and 3transmit to node 4 through a multiple access channel. Assume that thenoise in the broadcast channel is independent of that in the multipleaccess channel. The capacity region for the complete network cannot besolved by combining the capacity regions of the two component channels.For example, the maximal rate at which node 1 can send information tonode 4 is different when the broadcast channel is physically degradedthan when the broadcast channel is stochastically degraded even thoughthe capacity of the physically and stochastically degraded broadcastchannels are identical.

One special family of networks for which the network capacity isrelatively well-understood is the family of networks built fromnoiseless capacitated links (bit pipes). In these networks (bit-pipenetworks) under some very structured connection types, networkcapacities are precisely characterized. For example, for multicastconnections, where all messages start at a single node in the networkand each receiver wishes to reconstruct all messages transmitted by thatsource node, the capacity is fully characterized for all networktopologies. Unfortunately, finding capacities for general connectiontypes remains a difficult problem. For example, the capacity of anetwork with single multicast connection is solved, but the capacitiesof most networks with two or more multicast connections remain unsolved.

For the family of noiseless wireline networks under general connectiontypes, information theoretic inequalities can be used to find outerbounds on network capacities. The linear outer bound described in R. W.Yeung, “A framework for linear information inequalities,” IEEETransactions on Information Theory, 43(6): 1924-1934, November 1997 (thedisclosure of which is incorporated by reference herein in its entirety)gives the tightest outer bound implied by Shannon-type inequalities.This bounding technique has been implemented as the software programsInformation Theoretic Inequalities Prover (ITIP) and XITIP. The programhas complexity exponential in the number of links in the network and canthus only be used to compute capacity bounds for relatively smallproblem instances.

Since signals transmitted through real communication networks arecorrupted by physical phenomena such as noise and attenuation, the toolsfor computing capacities of bit-pipe networks cannot be directly appliedto compute capacities of real networks. As a result, the tools used toapproximate the capacities of real networks are quite different from thetools described above. The tools used to study large networks aretypically empirical, measuring achieved rate, available bandwidth, andbulk transfer capacity (BTC) across a single link or path. A networkmanager with administrative access to network routers and/or switchescan measure bandwidth metrics by gathering data rate statistics at allaccessible nodes (e.g., routers and switches). Since such access istypically available only to administrators and not to end users, endusers can often only estimate the bandwidth of individual links or pathsfrom end-to-end measurements. As a result, many such mechanisms yieldonly partial and unreliable assessments of a network's capacity.

SUMMARY OF THE INVENTION

Systems and methods are described for systematically computing truebounds on the performance of general networks under general connectiontypes. One embodiment includes a processor configured to obtain aninitial model of the communication network. In addition, the processoris further configured to perform at least one substitution that replacesa component of the initial network model with an equivalent or boundingmodel to produce a simplified network that is capable of beingcharacterized by the network analyzer, and the processor is alsoconfigured to characterize the initial model by characterizing thesimplified network model.

In a further embodiment, obtaining an initial model of the communicationnetwork comprises obtaining a definition of the nodes of the network, adefinition of the structure of the network, and a definition of theconnections within the network.

In another embodiment, the definition of the structure of the networkincludes the definition of at least one capacitated bit pipe.

In a still further embodiment, the definition of the structure of thenetwork includes the definition of at least one stochastic channel.

In still another embodiment, the processor is configured to factornetwork models by partitioning the network model into a series ofindependent network components.

In a yet further embodiment, the at least one substitution includessubstituting an equivalent or bounding network model for at least onestochastic component of the network model.

In yet another embodiment, the at least one stochastic component isreplaced with an equivalent or bounding bit-pipe model.

In a further embodiment again, the at least one substitution includessubstituting an equivalent or bounding network model for at least onedeterministic component of the network model.

In another embodiment again, the at least one substitution includesmodifying at least one bit-pipe component of the network model.

In a further additional embodiment, the processor is configured to boundthe accuracy of at least one of the models substituted for a componentof the network model to evaluate the impact of the substitution on theaccuracy with which the network analyzer can characterize the network.

In another additional embodiment, the processor is configured tocharacterize the simplified network model by outputting the simplifiednetwork model.

In a still yet further embodiment, the processor is configured tocharacterize the simplified network model by bounding a networkperformance characteristic of the simplified network model.

In still yet another embodiment, the network performance characteristicis a bound with respect to the capacity region of the simplified networkmodel.

In a still further embodiment again, the network performancecharacteristic is a bounding network on the capacity region of thesimplified network model.

In still another embodiment again, the network performancecharacteristic is a bound on the joint source-channel coding region ofthe simplified network model.

In a still further additional embodiment, the network performancecharacteristic is a bounding network on the joint source-channel codingregion of the simplified network model.

In still another additional embodiment, the processor is furtherconfigured to assess the bounding accuracy of the characterization ofthe simplified network model.

In a yet further embodiment again, the processor is configured to assessthe bounding accuracy of the characterization by comparing inner andouter bounds on the network performance.

In yet another embodiment again, the processor is configured to assessthe bounding accuracy of the characterization by comparing inner andouter bounding network models of the simplified network model.

In a further additional embodiment again, the processor is configured toperform the comparison on a link-by-link basis.

In another additional embodiment again, the processor is configured toperform the comparison on a cut-by-cut basis.

In a still yet further embodiment again, the processor is configured torepeat the at least one substitution using at least one alternativesubstitution in the event that the bounding accuracy of thecharacterization of the simplified network model is not sufficientlyaccurate.

In still yet another embodiment again, the processor is configured tofactor network models by partitioning the network into a series ofindependent network components, and the processor is configured torepeat the factoring of the network model and the at least onesubstitution using at least one alternative substitution in the eventthat at least one analysis criterion is not satisfied.

In a still yet further additional embodiment, the processor is furtherconfigured to perform the at least one substitution as part of arecursive substitution process in which components of the initialnetwork model are replaced with equivalent or bounding models to producea simplified network that is capable of being characterized by thenetwork analyzer.

In still yet another additional embodiment, the recursive substitutionprocess is a hierarchical substitution process.

Another further embodiment includes a processor configured to obtain aninitial model of the communication network. In addition, the processoris configured to factor the initial network model by partitioning thenetwork into a series of independent network components, and theprocessor is also configured to characterize the initial model bycharacterizing the simplified network model.

An embodiment of the method of the invention includes obtaining aninitial model of the communication network, performing at least onesubstitution that replaces components of the initial network model withequivalent or bounding models to produce a simplified network that iscapable of being characterized by the network analyzer, andcharacterizing the initial model by characterizing the simplifiednetwork model.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart illustrating a process for characterizing anetwork in accordance with an embodiment of the invention.

FIG. 2 is a flow chart illustrating a process for defining a network.

FIG. 3 is a flow chart illustrating a process for performing networksubstitution in accordance with an embodiment of the invention.

FIG. 4 is a flow chart illustrating a single iteration of a componentsubstitution process in accordance with an embodiment of the invention.

FIGS. 5a-5d illustrate various bit-pipe models.

FIG. 6 illustrates a generalized network topology including two sourcenodes and one sink node.

FIG. 7 illustrates a Y-shaped network topology include two source nodesand one sink node.

DETAILED DESCRIPTION

Turning now to the drawings, systems and methods for characterizingnetworks are disclosed. In several embodiments, a network analyzer(typically implemented via software on a computing device) analyzes oneor more performance characteristics of a network with respect to acollection of connections to be established across the network. Thenetwork analyzer can characterize the network in a variety of waysdepending upon the application for which the analysis is applied.Examples of performance characteristics that can be characterized by anetwork analyzer in accordance with embodiments of the invention include(but are not limited to) the capacity region of the network, the jointsource-channel coding region of the network, or the adversarial capacityof the network. Outputs of the network analyzer can take the form ofcomplete characterizations or responses to simpler queries such as “Canrates (R_(s):s∈

) for connections

be supported by the network” A detailed discussion of the performancecharacteristics that can be characterized by a network analyzer inaccordance with embodiments of the invention is provided below. In manyembodiments, the characterization involves providing an inner boundand/or an outer bound on one or more specific characteristics of thenetwork. An inner (outer) bound on a capacity or joint source-channelregion is a subset (superset) of that region. In other instances of theinvention, the network characterization takes the form of anothernetwork (a bounding network) that bounds a characteristic of theoriginal network. When the bounding network for the outer bound (theouter bounding network) and the bounding network for the inner bound(the inner bounding network) are identical, the bounding network isequivalent to the original network since its performance according tothe measure of interest (e.g., its capacity or joint-source channelcoding region) is identical to that of the original network. Networkcharacterization can also involve characterizing the accuracy of any ofthe bounds produced by the network analyzer.

In many embodiments, a network analyzer is not capable of directlycharacterizing an initial model of a network provided to the networkanalyzer. Therefore, a substitution process is performed to modify thenetwork so that it is capable of being characterized. The substitutionprocess involves replacing one or more components of the network withequivalent models or inner or outer bounding models (collectively,bounding models). If the inner bounding model and the outer boundingmodel for a particular component are identical, then that model is anequivalent model. If some replaced components are replaced by inner(outer) bounding models while any other replaced components are replacedby equivalent models, then the result is an inner (outer) boundingnetwork. If all components are replaced by equivalent models, then thenetwork is an equivalent network. The process can be performediteratively until the modifications result in a network model that iscapable of being analyzed by the network analyzer within an acceptableamount of time. The resulting network model is either equivalent to oran inner (outer) bound on the original network. In several embodiments,the network analyzer characterizes the network by outputting themodified network. As is discussed further below, the network analyzercan also characterize the modified network in a variety of waysincluding but not limited to providing one or more bounds on a specificcharacteristic of the network.

In a number of embodiments, a network analyzer characterizes a networkusing a sequence of processes that are described further below. Anetwork model is first entered into the analyzer. In some embodiments,the model is described by deterministic components with no stochasticelements (e.g., the network model only includes error-free capacitatedlinks). In other embodiments, the model is described by components withone or more stochastic elements. In other embodiments, the network isdescribed by a combination of stochastic and deterministic elements. Inseveral embodiments, the model can be factored into a collection ofindependent components. Factoring is the process of partitioning thenetwork into independent subnetworks. Combining the separate analysesthen enables an analysis of the entire network. In a number ofembodiments, a substitution process is applied to the network model inorder to obtain a modified network that is either equivalent to or is aninner or outer bound on the original network and is capable of beingcharacterized by the network analyzer. When a network model is obtainedthrough factoring and/or substitution that is capable of beingcharacterized, a network analyzer in accordance with embodiments of theinvention can bound a characteristic of the network. If the boundingnetwork produced by the substitution is an inner (outer) boundingnetwork, then an inner (outer) bounding algorithm is applied, yieldingan inner (outer) bound on the network performance characteristic of theoriginal network. If the bounding network is an equivalent network, thenthe network analyzer can apply an inner bounding algorithm to get aninner bound, apply an outer bounding algorithm to obtain an outer bound,or apply both to obtain both inner and outer bounds. In a number ofembodiments, the accuracy of the resulting bounds on the characteristicof the network (e.g., capacity or joint source-channel coding region) iscalculated. In several embodiments, accuracy is bounded by comparinginner and outer bounds. In other embodiments, accuracy is bounded bycomparing the bounding networks. Network analyzers and processes forcharacterizing networks, including processes for performingsubstitution, in accordance with embodiments of the invention arediscussed further below.

An Overview of Network Characterization Processes

A process for characterizing a network including various optionalprocesses that can be performed in accordance with embodiments of theinvention is illustrated in FIG. 1. The process 100 commences with thespecification (102) of a network model. In many embodiments, the networkmodel is provided to the network analyzer. In several embodiments, thenetwork analyzer provides a user interface enabling the specification ofthe network model . Network models and the manner in which they arespecified are discussed further below. Once a network model has beenobtained, the process can optionally involve factoring (104) the networkmodel. As is described below, factoring is the process of partitioningthe network into a series of independent subnetworks. In a number ofembodiments, factoring is unnecessary as the network model is entered bydescribing a collection of independent components and the nodes thatcommunicate through them. For many networks, some or all of the networkmodel cannot be factored. As is discussed further below, the portion ofthe network model that cannot be factored can be substituted with anetwork model that is either equivalent or is an inner or outer boundthat can be factored. When a substitution is performed to facilitatefactoring, the substitution can be referred to as a boundingfactorization.

A process involving the substitution of network components is thenperformed (106) to provide a network model that is either equivalent toor is an inner or outer bound on the original network and is capable ofbeing characterized by the network analyzer. In a number of embodiments,the substitutions are hierarchical. In other embodiments, any of avariety of strategies can be utilized to select substitutions toperform. The output of the network analyzer can be the network model,which can be analyzed manually, or using a separate network analysistool. The network model can also provide insight into the architectureof the current network and the design of future networks.

In many embodiments, the network analyzer characterizes the network(108) by determining relevant bounds (i.e., inner, outer, or inner andouter bounds) on one or more characteristics of the network. When thenetwork analyzer characterizes the network, the network analyzer canalso bound (110) the accuracy of the bounds. The accuracy of thecharacterization can be bounded using techniques including (but notlimited to) comparing inner and outer bounds, or comparing models (forexample, in some embodiments comparisons are made on a link-by-linkbasis; in others, comparisons are made on a cut-by-cut basis).

When the network is characterized, the process determines (112) whetherone or more predetermined analysis criteria are satisfied, indicatingthat the process can complete by outputting (114) a final networkcharacterization, In several embodiments, the network analysis criteriainclude (but are not limited) to criteria associated with the complexityof the network model, whether the network model can be further factored,the bounds and/or the accuracy of the bounds. Typically, the processiterates when factoring the network involves inserting a boundingfactorization, or the bounding accuracy does not satisfy a predeterminedcriterion. When the process iterates, the process can analyze thenetwork model and/or the characterization of the network model toidentify (116) components that are impacting the analysis criteria. Theprocess can then iteratively repeat using different factorizations(104), different sequences of substitutions (106), or both until theanalysis criteria are satisfied. At which point, the process outputs(114) the network characterization.

Although specific processes are illustrated in FIG. 1, any of a numberof processes can be utilized to characterize a network in accordancewith the requirements of a specific application. The various processesshown in FIG. 1 involve the use of a network model. The manner in whicha network can be described using a network model and the manner in whichthe various processes shown in FIG. 1 operate on the network model arediscussed further below.

Definition: Network Models

The nature of the network model definition used by a network analyzer inaccordance with an embodiment of the invention typically is dependentupon the nature of a specific application. Two examples of approaches tothe modeling of communication networks are the graph-theoretic approach,which treats networks as graphs including nodes connected by noise-free,capacitated links that can be used error free up to a certain capacity,and the information theoretic approach, which treats networks asstochastic relationships between the input and outputs variables at acollection of nodes. Models used in embodiments of the invention includebut are not limited to the example model types described below.

In some embodiments, a network analyzer models a network as a directedgraph

=(

,

), where

and

⊂

×

denote the set of nodes and links, respectively. Each directed linke=(v₁, v₂)∈

represents a noiseless link (bit pipe) of capacity C_(e) from node v₁ tonode v₂ in

. For each node v, In(v) and Out(v) denote the set of incoming andoutgoing links of node v. For each edge e, Start(e) and End(e) denotethe input and output nodes for edge e.

Noiseless capacitated links are typically employed to model cables orwires in wireline networks. In real networks, signal transmissions overcables and wires suffer from various forms of signal degradation and/orcorruption by noise. Wireline channels are therefore more accuratelymodeled using a distribution that describes the statistical relationshipbetween the transmitted and received signals. In some embodiments, anetwork analyzer models a network as a directed graph

=(

,

), where directed link e=(v₁,v₂)∈

represents a noisy channel with conditional distributionp_(e)(y_(e)|x_(e)) describing the distribution on the channel's outputy_(e) at node v₂ when the channel's input at node v₁ is x_(e).

In some embodiments, a network analyzer models a network as a directedgraph

=(

,

₁∪

₂) where each e∈

₁ describes a noiseless capacitated link of capacity C_(e) and each e∈

₂ describes a noisy channel with conditional distributionp_(e)(y_(e)|x_(e)).

In wireless networks, signals are transmitted directly into a node'ssurrounding environment (e.g., air, water, space) rather than beingtransmitted over a cable or wire. Wireless transmissions suffer signaldegradation and/or corruption by noise. In addition, signals broadcastinto their surrounding environment may be overheard by communicationdevices other than their intended receiver(s) and/or may sufferinterference caused by distinct signals intended for the same receiveror other receivers in the network. Such phenomona do not readily lendthemselves to direct modeling using graph theoretic approaches.Broadcast, multiple access, and interference channel models are allexamples of simple channel models used to model the phenomena associatedwith wireless transmissions. All of these channel types are instances ofinformation theoretic multiterminal channel models. In a general memoryless multiterminal channel model with m inputs and n outputs, a networkis modeled by: (1) a collection

of channel nodes, (2) a pair (v_(x), v_(y)) of correspondences such thatfor each i∈{1, . . . , m} and j∈{1, . . . , n} v_(x)(i)∈

describes the node transmitting network input x_(i) and v_(y)(j)∈

describes the node receiving network output y_(j), and (3) adistribution p(y₁,y₂, . . . , y_(n)|x₁,x₂, . . . , x_(m)) describing theconditional distribution on the network outputs given the networkinputs. When the channel has memory, then the distribution on channeloutputs depends on current and past channel inputs. The channel inputsand outputs may be scalars or vectors. Multiple transmissions from asingle node (v_(x)(i₁)=v_(x)(i₂) for some i₁≠i₂, i₁,i₂∈{1, . . . , m})and/or multiple receptions at a single node (v_(y)(j₁)=v_(y)(j₂) forsome j₁≠j₂, j₁,j₂∈{1, . . . , n}) are employed in some instances of thisinvention. General multiterminal channel models are employed in manyinstances of this invention.

A network analyzer in accordance with embodiments of the invention canbuild an initial model of a network using well-understood processesbased upon distribution estimation. For the purposes of networkinformation theory, these processes model the stochastic relationship ofinput and output variables at nodes in the network. In some embodiments,these stochastic relationships capture network features including (butnot limited) the statistics of the noise, the statistical dependence, ifany, of the noise on the transmitted signal or signals, interference (ifany) between transmitted signals, signal attenuation, and multipathfading characteristics. In many embodiments, these stochasticrelationships are estimated by observing the statistics of networkinputs and outputs at nodes in a network. In several embodiments, thenetwork analyzer obtains network statistics in real time from a networkadministration application. In other embodiments, the network analyzerprocesses collected data offline. In several embodiments, the networkanalyzer employs components from a library of components as connected bya network designer.

Many communication networks employ both wireless and wirelinetransmissions. For example, while much of the core of the internet is awireline network, that network is frequently accessed with wirelessdevices like laptops, mobile phones, and PDAs. Conversely, while manysensor networks employ wireless technology at the sensors, the collecteddata is often processed and shared over wireline networks. Toaccommodate such hybrid networks, in some instances of the invention,the network model includes both multiterminal channels and (noisy and/ornoiseless) wireline links.

In some instances, portions of the network model are entered into thenetwork analyzer by describing the vertices and directed edges of agraph. In several instances, portions of the network model are enteredinto the network analyzer by describing a collection of vertices, one ormore distributional models, and the correspondences describing theirinputs and outputs. In many instances, the network model is entered intothe network analyzer using a graphical user interface.

A process for describing a network model in accordance with anembodiment of the invention is illustrated in FIG. 2. The process 120involves defining (122) the nodes within the network, which representthe individual transmitters and receivers within the network. Theprocess then involves defining (124) the links and/or channels withinthe network. The links refer to the error-free capacitated links withinthe network, and the channels refer to the stochastic and/ormultiterminal channels within the network. Once the nodes, and thelinks/channels within the network are defined, the demands on thenetwork (i.e., the desired communication connections within the network)are defined (126). The combination of the nodes, the structure of thenetwork, and the connections within the network form a model of anetwork that can be characterized by a network analyzer in accordancewith embodiments of the invention. The manner in which connections canbe defined within a network model in accordance with embodiments of theinvention is described below.

Definition: Network Sources and Connections

The network analyzer analyzes network performance with respect to acollection of connections to be established across the network. Theconnections are described by a connection set

. Each connection s=(v,

)∈

describes a single transmitter v∈

and one or more receivers

⊂

\{v}. The network source is a vector U=(U_(s):s∈

) of |

| source random variables defined by a source distribution p.Distribution p may describe a memoryless random process or a randomprocess with memory. The nature of the sources and connections inaccordance with an embodiment of the invention varies with theapplication. Examples of connection types consistent with embodiments ofthe invention include but are not limited to the following examples.

In some instances of the invention, the connections are losslessconnections and the sources are independent. Lossless connections andindependent sources are used, for example, in characterizing thecapacity region of a network

under connections

. Specifically, the capacity region

(

,

) is the set of all rate vectors (R_(s):s∈

) such that when (U_(s):s∈

) is uniformly distributed on Π_(s∈)

{1, . . . , 2^(nR) ^(s) }, it is possible to simultaneously and reliablydeliver each source U_(s)=U_((v,)

₎ from node v to all of the receivers in

using n channel uses. In some instances of the invention, reliableinformation delivery means that the error probability of the informationreconstruction at each receiver can be made arbitrarily small as theblocklength n grows without bound. In other instances of the invention,reliable information delivery means that the error probability of theinformation reconstruction at each receiver can be made precisely equalto zero for some n sufficiently large.

In many instances of the invention, the connections may allow imperfect(lossy) data reconstruction and/or the sources may be statisticallydependent. Lossy data reconstruction and dependent sources are used, forexample, in characterizing the optimal joint source-channel codingregion of a network

under a source distribution p and connections

. The joint source-channel coding region

(

, p,

) is the set of all distortion vectors (D_(s,t):s=(v,

)∈

,t∈

) such that for each connection s=(v,

)∈

and each receiver t∈

, node t can reconstruct source U_(s) with expected distortion nogreater than D_(s,t). The distortion measures used in the distortionconstraint D_(s,t) for each connection s and corresponding receiver tvary with the application to which the network analyzer is applied. Insome instances, the distortion measures are identical for allconnections and all receivers. In other instances, the distortionmeasure varies from connection to connection but is fixed for eachconnection. In several instances, the distortion measure for someconnections varies among the receivers. For example, in someembodiments, the distortion constraints for one connection are specifiedwith respect to the squared error distortion measure while thedistortion constraints for another connection are specified with respectto the Hamming distortion measure. In many embodiments, the distortionconstraints for one or more connections s=(v,

)∈

involve the reconstruction of source U_(s) at receiver t∈

have expected distortion no greater than D_(s,t) with respect to theabsolute error distortion measure while the reconstruction of sourceU_(s) at receiver t′∈

has expected distortion no greater than D_(s,t′) with respect to theHamming distortion measure. In some instances of the invention, one ormore connections in the network represent functional demands. Theseinstances are accommodated in the above framework by appropriate networkdefinitions. For example, if s₁=(v₁, T₁) and s₂=(v₂, T₂) are two networkconnections (s₁, s₂∈

) with receiver set sizes |T₁|=|T₂|≧0 and node v₂ wishes to reconstructthe function f(U_(s) ₁ , U_(s) ₂ ), defining a node v₄ to serve astransmitter for connection s₃=(v₄, {v₃}) with source U_(s) ₃ =f(U_(s) ₁,U_(s) ₂ ) implements the desired functional demand. If node v₄ is notconnected to the rest of the network, then the reconstruction of sourceU_(s) ₃ must be built using only descriptions from connected nodes.

Although specific examples of network models and connection types aredescribed above, any of a variety of network models, node definitions,channel definitions, and connection definitions can be utilized inspecifying a model of a network for characterization by a networkanalyzer in accordance with an embodiment of the invention. Beforediscussing the manner in which network analyzers in accordance withembodiments of the invention can characterize a network described usinga network model similar to the network models described above, adiscussion is provided below of various ways in which a network can becharacterized.

Definition: Network Characterization

The network characterization that is the output of the network analyzervaries with the application for which the analysis is applied. Inseveral embodiments, the network analyzer characterizes the networkusing a bound on the performance of a network

with respect to a given connection set

and source distribution p. For example, in some embodiments of theinvention the network characterization is an inner or outer bound on thenetwork capacity

(

,

), or a bound on some function of the values in that region, e.g., abound on the Lagragian max_((R) _(s) _(:s∈)

_()∈)

₍

_(,Θ))λ_(s)R_(s) for some fixed positive constants (λ_(s):s∈

). In other embodiments of the invention, the network characterizationis an equivalent or bounding network with respect to a desiredperformance metric, as defined below. In many embodiments, thisequivalent or bounding network is simpler than the original network. Forexample, the bounding network is an equivalent or bounding network andhas no stochastic components; or the bounding network has fewer nodesand/or edges and bounds the performance of the original network withrespect to a given connection set

.

Given a performance metric, a network

′ is equivalent to input network

under connection set

if the performance of

′ is identical to the performance of

under connection set

. For example, network

′ is equivalent to

with respect to capacity on connection set

if

(

,Θ)=

(

′,Θ). In many but not all instances of the network analyzer, theanalyzer employs models that are equivalent for all possible connectionsets (

(

,

)=

(

′,

) for all

). Similarly, network

′ is an inner bounding network for

on connection set

if the set of achievable performances of

′ is a subset of that of

over a given set of connections

and sources (U_(s):s∈

) (e.g.,

(

′,

)⊂

(

,

). In many but not all instances of the network analyzer, the analyzeremploys bounding models that apply to all sources and connections. Anetwork

′ is an outer bounding network

for on connections

and sources (U_(s):s∈

) if

is an inner bounding network for

′ on the given connections and sources.

Network Characterization Processes

Referring back to FIG. 1, once a network model defined in a mannersimilar to the various network models outlined above has been obtained(102) a process similar to the process 100 illustrated in FIG. 1 can beutilized to characterize the network. As was explained above, in manyembodiments, the process of characterizing a network involves factoringthe network model prior to attempting to characterize the model.Processes for factoring network models in accordance with embodiments ofthe invention are discussed below.

Factoring Networks

In many embodiments of the invention, where part or all of the networkmodel is a distributional model, the distributional model is factoredinto smaller or simpler components. In some embodiments, this factoringalgorithm precisely factors the network independent sub-networks.Precisely factoring a distributional model p(y₁, . . . , y_(n)|x₁, . . ., x_(m)) of a network or sub-network involves finding a component set

, a pair of partitions

=(P_(e):e∈

) and Q=(Q₃:e∈

) for {1, . . . , m} and {1, . . . , n}, respectively, and a collectionof conditional distributions {p_(e)(y_(e)|x_(e)):e∈

} for which

p(y₁, …  , y_(n)|x₁, …  , x_(m)) = p_(e)(y_(e)|x_(e)).

Here x_(e)=(x_(i):i∈P_(e)) and y_(e)=(y_(j):j∈Q_(e)) for each e∈

, and a partition (A₁, . . . A_(k)) of a set A is a collection ofsubsets of A(A_(i) ⊂A) that do not intersect (A_(i)∩A_(j)=for all i≠j)and together contain all of the elements of A(∪_(i=1) ^(k)A_(i)=A).

In other embodiments of the invention, the factoring algorithm involvesfinding a factored distributional model that is a bounding model for theoriginal network. For example, given a network

which is characterized in part or in whole by a distributional modelp(y₁, . . . , y_(n)|x₁, . . . , x_(m)), in some instances of theinvention, the factoring algorithm finds partitions

=(P_(e):e∈

) for {1, . . . , m} and Q=(Q_(e):e∈

) for {1, . . . , n} and conditional distributions p_(e)(y_(e)|x_(e))for all e∈

such that the network

′ that results when characterization p(y₁, . . . , y_(n)|x₁, . . . ,x_(m)) is replaced by the factored characterization

p^(′)(y₁, …  , y_(n)|x₁, …  , x_(m)) = p_(e)(y_(e)|x_(e))

gives an inner (outer) bounding network. That is,

(

′,

)⊂

(

,

)(

(

′,

)⊃

(

,

)) for all

or for a particular

of interest in instances that calculate capacity or

(

′,p,

)⊂

(

,p,

)(

(

′,p,

)⊃

(

,p,

)) for all p and

or for a particular (p,

) of interest in instances that calculate the joint source-channelcoding region. For example, the paper M. Effros, “On dependence anddelay: capacity bounds for wireless networks,” IEEE WirelessCommunications and Networking Conference, Paris, France, April 2012 (toappear), (the disclosure of which is hereby incorporated by reference inits entirety), shows that for any distributional channel model p(y₁, . .. , y_(n)|x₁, . . . , x_(m)) for which

p(y ₁ , . . . , y _(l) |x ₁ , . . . , x _(k) , x _(k+1) , . . . , x_(m))=p(y ₁ , . . . , y _(l) |x ₁ , . . . , x _(k))

p(y _(l+1) , . . . , y _(n) |x ₁ , . . . , x _(k) , x _(k+1) , . . . , x_(m))=p(y _(l+1) , . . . , y _(n) |x _(k+1) , . . . , x _(m)),

the factored distributional model

p(y ₁ , . . . , y _(l) |x ₁ , . . . , x _(k))p(y _(l+1) , . . . , y _(n)|x _(k+1) , . . . , x _(m))

is a lower bounding model for p(y₁, . . . ,y_(n)|x₁, . . . , x_(m)) fromthe perspective of capacity for all

and also a lower bounding model for p(y₁, . . . , y_(n)|x₁, . . . ,x_(m)) from the perspective of joint source-channel coding for all (p,

).

In some embodiments, factoring is accomplished by fixing an inputdistribution and then running a distribution factoring algorithm on theresulting joint distribution. In several embodiments, the inputdistribution for each finite-alphabet input is the uniform distribution.In a number of embodiments, the input distribution for each real-valuedinput is an independent Gaussian distribution that satisfies all powerconstraints on that input. In certain embodiments, multiple inputdistributions are employed. A variety of processes can be used to factora distribution. For example, Algorithm 3.5, “Finding the class PDAGcharacterizing the P-map of a distribution P” in the text bookProbabilistic Graphical Models by D. Roller and N. Friedman is oneexample of an algorithm used for factoring distributions (the disclosureof which is incorporated by reference herein in its entirety). In otherembodiments, any processes for factoring a network model appropriate toa specific network model and/or application can be utilized inaccordance with embodiments of the invention.

In a number of embodiments, the factoring algorithm is a hierarchicalfactoring algorithm that first factors a distributional component of thenetwork into sub-components and then applies the factoring algorithm toone or more of those sub-components. In some embodiments, the factoringalgorithm is ran as a parallel algorithm, with simultaneous applicationof the factoring algorithm to multiple components run in parallel eitheron distinct processors of a single device or on distinct devices.

In many processes in accordance with embodiments of the invention, thedistribution factoring algorithm employs conditional mutual informationqueries. For example, consider a model in which the alphabets of allnetwork inputs X₁, . . . , X_(m) and all network outputs Y₁, . . . ,Y_(n) are finite. Let A and B be subsets of {1, . . . , m} and {1, . . ., n}, respectively, and let X_(A)=(X_(i):i∈A), X_(A) _(c) =(X_(i):i∈{1,. . . , m}\A), Y_(B)=(Y_(j):j∈B), and Y_(B) _(c) =(Y_(j):j∈{1, . . . ,n}\B). Notice that I(X_(A) _(c) ;Y_(B)|X_(A))=0 under the uniformdistribution on (X₁, . . . , X_(m)) if and only if I(X_(A) _(c) ;Y_(B)|X_(A)=x_(A))=0 for all possible values x_(A) of X_(A). ThereforeI(X_(A) _(c) ;Y_(B)|X_(A))=0 under the uniform distribution on (X₁, . .. , X_(m)) implies the conditional independence of X_(A) _(c) and Y_(B)given X_(A) for all possible distributions p(x₁, . . . , x_(m)) on thecomponent inputs. As a result, for any pair of sets A⊂{1, . . . , m} andB⊂{1, . . . , n} for which

I(X _(A) _(c) ;Y _(B) X _(A))+I(X _(A) ;Y _(B) _(c) )+I(Y _(B) ;Y _(B)_(c) |X _(A) ,X _(A) _(c) )=0,

the distribution p(y₁, . . . , y_(n)|x₁, . . . , x_(m)) can be factoredas

p(y ₁ , . . . , y _(n) |x ₁ , . . . , x _(m))=p(y _(B) |x _(A))p(y _(B)_(c) x _(A) _(c) ).

In several embodiments, finding such a set pair (A,B) and factoring thenetwork into two components is one step in a factoring algorithm.

In a number of embodiments, applying the test

“Is I(X_(A) _(c) ;Y_(B)|X_(A))+I(X_(A);Y_(B) _(c) |X_(A) _(c))+I(Y_(B);Y_(B) _(c) |X_(A),X_(A) _(c) ) equal to zero!”

to all candidate sets A⊂{1, . . . , m} and B⊂{1, . . . , n} with |A|≦k₁and |B|≦k₂ finds all independent channels with at most k₁ inputs and atmost k₂ outputs in the network. When k₁ and k₂ are finite, the number ofpossible sets A and B of the given sizes is polynomial in |

|.

In some instances of the invention, where distributional model p(y₁, . .. , y_(n)|x₁, . . . , x_(m)) is known to be built from independentchannels each of which has at most k₁ inputs, the factorization

p(y₁, …  , y_(n)|x₁, …  , x_(m)) = p(y_(e)|x_(e))

is found as follows. Choose an input distribution

${p\left( {x_{1},\ldots \mspace{14mu},x_{m}} \right)} = {\prod\limits_{i = 1}^{m}{{p\left( x_{i} \right)}.}}$

For each j∈{1, . . . , n}, find the set

⊂{1, . . . , m} defined as

$_{j} = {\arg \; {\min\limits_{A:{{A} \leq k_{1}}}{{I\left( {X_{A^{c}};\left. Y_{j} \middle| X_{A} \right.} \right)}.}}}$

Since the inputs are drawn from an independent distribution, thefunction f(A)=I(X_(A) _(c) ;Y_(j)|X_(A)) is a submodular function, asshown in the paper T. S. Han, “The capacity region of generalmultiple-access channel with certain correlated sources,” Informationand Control, vol. 40, pp. 3760, January 1979, the disclosure of which isincluded by reference in its entirety. Hence, the solution of thedescribed optimization problem can be found in polynomial time usingmethods known in sub-modular optimization. In some instances of theinvention, the sub modular optimization algorithm described in the paperA. Schrijver, “A combinatorial algorithm minimizing submodular functionsin strongly polynomial time,” Journal of Combinatorial Theory, vol. 80,pp. 346355, 2000, is employed in this optimization. The disclosure ofthis paper is hereby included by reference in its entirety. Afterfinding the sets

_(j) corresponding to each j, in some instances of the invention, thepair of partitions (P_(e):e∈

) and (Q_(e):e∈

) is formed by searching for the connected components in the bipartitegraph constructed by connecting each j∈{1, . . . , n} to each i∈

_(j).

In some instances of the invention, the network's stochastic componentis described not by a distributional model p(y₁, . . . , y_(n)|x₁, . . ., x_(n)) but instead by r samples

(x_(1,t), . . . , x_(m,t), y_(1,t), . . . , y_(n,t)), t∈{1, 2, . . . r},

drawn i.i.d. from distribution

${\prod\limits_{i = 1}^{m}{{p\left( x_{i} \right)}{p\left( {y_{1},\ldots \mspace{14mu},\left. y_{n} \middle| x_{1} \right.,\ldots \mspace{14mu},x_{m}} \right)}}},$

where Π_(i=1) ^(m)p(x_(i)) is an independent input distribution andp(y₁, . . . , y_(n)|x₁, . . . , x_(m)) is the unknown channeldistribution. In some instances of the given invention, the user inputsa bound k₁ on the maximal number of inputs per component (k₁≦m) and thefactoring algorithm employs a model

selection algorithm to find a factored model

p(y_(e)|x_(e))

with at most k₁ inputs per component (k₁≦m) for the unknown distributionp(y₁, . . . , y_(n)|x₁, . . . , x_(m)). The aim of the model selectionalgorithm is to select a model that fits the observed data withoutunder-fitting or over-fitting. In some instances of the invention, thismodel selection process is accomplished using the well-known modelselection methods of Bayesian or Schwarz information criterion (BIC)described in the paper Shwarz, G., “Estimating the dimension of amodel,” Ann. Statist., 6, 461-464, 1978, the disclosure of which isincorporated by reference in its entirety. According to BIC, for eachj∈{1, . . . , n},

_(j) is set to the set A that minimizes

H ^  ( Y j | X A ) + ( log   r 2  r )  (  j  - 1 )  ∏ i ∈ A   i  ,

over all A∈{1, . . . , m}, such that |A|≦k₁. For A⊂{1, . . . , m} andj∈{1, . . . , n}, empirical conditional entropy Ĥ(Y_(j)|X_(A)) isdefined as

H ^  ( Y j | X A ) = - ∑ y j ∈ j , x A ∈ ∏ i ∈ A  X i  p ^  ( x A ,y j )  log   p ^  ( y j | x A ) ,

and empirical conditional distribution {circumflex over(p)}(y_(j)|x_(A)) of y_(j) given x_(A) is defined as

${{\hat{p}\left( y_{j} \middle| x_{A} \right)} = \frac{\hat{p}\left( {x_{A},y_{j}} \right)}{\hat{p}\left( x_{A} \right)}},{where}$${{\hat{p}\left( {x_{A},y_{j}} \right)} = \frac{\left\{ {{{t\text{:}\left( {x_{A,t},y_{j,t}} \right)} = \left( {x_{A},y_{t}} \right)},{t = 1},\ldots \mspace{14mu},r} \right\} }{r}},{and}$${\hat{p}\left( x_{A} \right)} = {\frac{\left\{ {{{t\text{:}x_{A,t}} = x_{A}},{t = 1},\ldots \mspace{14mu},r} \right\} }{r}.}$

Unlike function f(A), function

g  ( A ) = H ^  ( Y j | X A ) + ( 2  r ) - 1  ( log   r )  (  j - 1 )  ∏ i ∈ A    i 

is not a submodular function. However, in the case where |

₁|= . . . =|

_(m)|

|

|, the solution of this optimization can still be found in polynomialtime using the following procedure. For each k∈{1, . . . , k₁}, find theset A_(k) such that

$A_{k} = {\arg \; {\min\limits_{{A} \leq k}{{\hat{H}\left( Y_{j} \middle| X_{A} \right)}.}}}$

After performing these k₁ different optimizations, let

 j = arg   min A ∈ { A 1 , …  , A k 1 }  [ H ^  ( Y j | X A ) + (log   r 2  r )  (  j  - 1 )      A  ] .

Each of the k₁ optimizations required to find A₁, . . . , A_(k) ₁consists of minimizing a submodular function subject to a modularconstraint, and hence can be done efficiently in polynomial time.

Although specific factoring algorithms are discussed above, any of avariety processes appropriate to specific applications can be utilizedfor factoring of networks in accordance with embodiments of theinvention.

Component Modification

In some instances of the network analyzer, part or all of the network (anetwork component) is replaced by an inner or outer bounding model withrespect to the sources and connections of interest to the networkcomponent. If the inner and outer bounding model for a component areidentical, then the component model is equivalent to the originalcomponent. Where calculation of network performance using currentbounding network model(s) is too computationally intensive, the networkanalyzer can recursively simplify the bounding model by replacingcomponents of the network with simpler bounding models. The recursioncan proceed until the replacements yield a sufficiently simple boundingmodel as measured by a simplicity criterion, which is typicallydependent on the application. Measures of simplicity may include (butare not limited to) numbers of nodes and/or edges in the network and/ortemporal requirements on the run-time for the network characterizationprocedure. In some instances, this procedure for modifying the networkfacilitates or employs the application of computational bounding toolsby the network analyzer.

Component Modification Processes

A process for modifying a network model to obtain a model that iscapable of being characterized in accordance with an embodiment of theinvention is illustrated in FIG. 3. The process 130 includes attemptingto model (132) each of the components in the network with an equivalentmodel or an inner or outer bounding model with respect to the sourcesand connections associated with the component. Once a complete pass ismade over the network model, the modified network model is output (134).In many embodiments, the modified network model is evaluated (136) todetermine whether it is capable of being characterized by a networkanalyzer. In the event that the network model does not satisfy one ormore predetermined simplicity criteria, the process iterates until thecriteria are satisfied.

Although a specific component modification process is illustrated inFIG. 3, any of a variety of network modification processes capable ofoutputting a model that is capable of being characterized by a networkanalyzer in accordance with embodiments of the invention can beutilized. Specific component modification processes in accordance withembodiments of the invention are discussed further below.

Component Modification Processes

Some methods for component modification can be applied to bothstochastic components and bit-pipe components while others applyspecifically to one component type or the other. A process for modifyingcomponents of a network model in accordance with an embodiment of theinvention is illustrated in FIG. 4. The process 140 includes one or moreof the following procedures.

In some instances of the invention, process 140 employs a procedure tosubstitute (142) equivalent networks, or inner (outer) bounding networksfor stochastic components within the network model. The substitution caninvolve replacing a stochastic channel with either an equivalent networkmodel or a bounding model that can be (but is not limited) a bit-pipemodel. There are a number of ways in which the substitution can beachieved. In many instances, the network component can be looked up in alibrary of component substitutions. In other instances, a network modelthat is equivalent or bounding can be found by trial and error or as asolution to a collection of characterizing equations. In some instancesof the invention, this solution is an optimized solution. Varioustechniques for obtaining equivalent or bounding network models inaccordance with embodiments of the invention are discussed below.

In some instances of the invention, the process 140 includes a procedureto modify (144) the components of the network. The modification of thecomponents of the network typically involves modifying the nodes and/orthe capacitated links in the network to facilitate the characterizationof the network.

In several embodiments, component modification processes involveadditional operations involving the bounding (146) of the accuracy ofthe network models substituted for each of the components. Although notutilized in all instances, bounding (146) the accuracy of the modelssubstituted for network components can aid in the identification ofsubstitutions that are likely to reduce the accuracy of the overallcharacterization of the network.

In some instances of the invention, when accuracy bounds are used, theprocess can evaluate (148) whether specific bounding models aresufficiently accurate or whether alternative bounding models should beutilized. In which case, the process can iterate until a satisfactoryset of modifications has been made. Various component modifications thatcan be applied to stochastic components and/or bit-pipe components inaccordance with embodiments of the invention are discussed furtherbelow.

In many embodiments, the accuracy of component modifications can beassessed by comparing the inner and outer bounding models for a givencomponent. Methods used for such comparisons are described furtherbelow. In several embodiments, the manner in which components arereplaced is at least partially determined by the assessed accuracy ofthe inner and outer bounding models. In a number of embodiments,processes are utilized that seek to choose components that yield thetightest inner and outer bounding models.

Although specific processes are described below for modification ofcomponents, any of a variety processes appropriate to specificapplications can be utilized to modify network components in accordancewith embodiments of the invention.

Component Modification: Merging and Removing Nodes

One of the simplest operations for deriving an outer bounding network ismerging nodes. Coalescing two nodes is equivalent to adding two links ofinfinite capacity—one from the first node to the second, and anotherfrom the second node to the first—thereby guaranteeing an outer boundingnetwork under all connection types. When a pair of nodes, say v₁ and v₂,are combined to form a new node v₃, all outgoing connections (i.e., all(v₁/

)∈

and/or (v₂,

₂)∈

) and incoming connections (i.e., all (v,

)∈

for which

∩{v₁, v₂}≠) are reassigned to v₃. Thus

′ replaces any (v₁,

₁)∈

by (v₃,

₁)∈Υ′ and (v₂,

₂)∈

by (v₃,

₂)∈

′and replaces any (v,

)∈

with

∩{v₁,v₂}≠ by (v,

′) where

′=((

\{v₁,v₂})∪{v₃}). Transmission and reception of channel inputs andoutputs are likewise transferred from the prior separated nodes to thenew combined node. For example, for any i such that v_(x)(i) fell in set{v₁,v₂} in the original network, v_(x)(i) is set to v₃ in the modifiednetwork. Likewise, for any j such that v_(y)(j) fell in set {v₁,v₂} inthe original network, v_(y)(j) is set to v₃ in the modified network.

In some cases, nodes can be combined without affecting the networkcapacity. One simple example is when the sum of the capacities of theincoming links of a node v is less than the capacity of each of itsoutgoing links. In this situation, node v can be combined with all nodesw for which (v,w)∈

and node w has no other incoming links. Some examples of simpleoperations for deriving inner bounds include removing entire components,removing one or more channel inputs from any channel for which the inputalphabet has a “no transmission” symbol, and removing channel outputs.

Component Modification: Stochastic Components

In many embodiments, the network analyzer replaces one or moreindependent stochastic components (channels) by corresponding boundingmodels. The models may take the form of stochastic models, or networksof bit pipes, or any combination of stochastic and deterministiccomponents. From the perspective of capacity, if each model used in areplacement is both an inner bounding model and an outer bounding modelfor the channel that it replaces, then replacing those channels by theirmodels yields an equivalent network.

In a number of embodiments, the bounding models used to determine boundson network capacity only involve error-free capacitated links (losslessbit pipes). A channel model made of one or more lossless bit pipes iscalled a bit-pipe model. For the capacity problem, Koetter et al. haveestablished that an arbitrary collection of network connections can bemet on a network of noisy, independent, memoryless links if and only ifit can be met on another network where each noisy link is replaced by anoiseless bit pipe with capacity equal to the noisy link capacity. Thus,a bit pipe is an equivalent model for a noisy link of the same capacityfor ail possible connections

. Likewise, a noisy link is an equivalent model for any other noisy linkof the same capacity for all possible connections

. This equivalence between noisy, memoryless links and noiseless bitpipes of the same capacity is established in the papers: R. Koetter, M.Effros, and M. Medard, “On a theory of network equivalence.” Proceedingsof the IEEE Information Theory Workshop, pages 326-330, Volos, Greece,June 2009; R. Koetter, M. Effros, and M. Mèdard, “A theory of networkequivalence—Part I: Point-to-point channels.” IEEE Transactions onInformation Theory, pages 972-995, February 2011. The disclosure of thetwo papers by Koetter et al. is incorporated by reference herein in itsentirety. Also for the capacity problem, Koetter et al. have establishedinner and outer bounding models for multiterminal channels that apply toall connection sets

. Koetter et al. give both general methods for deriving bounding modelsand examples of the results of those methods for broadcast, multipleaccess, and interference channels. Both the methods and the examples aredescribed in the papers: R. Koetter, M, Effros, and M. Mèdard, “Beyondnetwork equivalence.” Allerton Annual Conference on Communications,Control, and Computing, Monticello, Ill., September 2010; R. Koetter, M.Effros, and M. Mèdard, “A theory of network equivalence, parts i andii.” ArXiv e-prints, July 2010. The disclosure of the two Koetter et al.papers is incorporated by reference herein in its entirety. Koetter etal. further propose developing bit-pipe models for a broad library ofnetwork components in such a way that the capacity of any networkcontaining the given stochastic components can be bounded by thecapacity of a network of noiseless bit pipes achieved by replacing eachstochastic component by its corresponding bit-pipe model. The resultsdemonstrate that for any known point in the capacity region (that is,for any achievability result) on any memoryless channel there exists acorresponding inner bounding bit-pipe model with topology and linkcapacities matched to the given point. This establishes both a largecollection of bit-pipe models and a methodology for deriving bit-pipemodels for new channels. The following are examples of bit-pipe modelsfor the capacity problem.

Exemplary Bit Pipe Models

A number of bit-pipe models for the capacity problem are discussed belowwith reference to the bit-pipe models illustrated in FIGS. 5a -5 d.Given any memoryless point-to-point channel (

, p(y|x),

), the bit pipe shown in FIG. 5a of capacity

$C = {\max\limits_{p{(x)}}{I\left( {X;Y} \right)}}$

is an equivalent bit-pipe model.

Consider a memoryless broadcast channel,

=(

,p(y₁,y₂|x),

×

). If

is a physically degraded broadcast channel of the formp(y₁,y₂|x)=p(y₁|x)p(y₂|y₁) or a stochastically degraded broadcastchannel, so that there exists a pair of conditional distributionsp′(y₁,y₂|x) and p′(y₂|y₁) such that p′(y₁,y₂|x)=p(y₁|x)p′(y₂|y₁) andp′(y₂|x)=p(y₂|x) for all (x,y₂), then let

={(R₁,R₂,R₁₂): for some p(u)p(x|u)p(y₁,y₂|x) R₁≦I(X;Y₁|U),R₂+R₁₂≦I(U;Y₂)}. Then for any (R₁,R₂,R₁₂)∈

the bit-pipe model shown in FIG. 5b is an inner bounding bit-pipe modelfor

. For any broadcast channel C=(

,p(y₁,y₂|x),

×

) (here no degradedness assumptions are employed), let

$_{1} = \left\{ {{{\left( {R_{1},R_{2},R_{12}} \right)\text{:}R_{12}} \geq {\max\limits_{p{(x)}}{I\left( {X;Y_{2}} \right)}}},{{R_{1} + R_{12}} \geq {\max\limits_{p{(x)}}{I\left( {{X;Y_{1}},Y_{2}} \right)}}}} \right\}$$_{2} = {\left\{ {{{\left( {R_{1},R_{2},R_{12}} \right)\text{:}R_{12}} \geq {\max\limits_{p{(x)}}{I\left( {X;Y_{1}} \right)}}},{{R_{2} + R_{12}} \geq {\max\limits_{p{(x)}}{I\left( {{X;Y_{1}},Y_{2}} \right)}}}} \right\}.}$

Then for any point (R₁,R₂,R₁₂)∈

, the bit-pipe model shown in FIG. 5b is an outer bounding bit-pipemodel for

. (For any set A, Ā designates the convex closure of A.)

Consider a memoryless multiple access channel,

=(

₁×

₂,p(y|x₁,x₂),

). Let

={(R ₀ ,R ₁ ,R ₂):R ₁ ≦I(X ₁ ;Y|X ₂), R ₂ ≦I(X ₂ ;Y|X ₁), R ₁ +R ₂ ≦I(X₁ ,X ₂ ;Y), R₀=0 for some p(x₁)p(x₂)}

₁={(R ₀ ,R ₁ ,R ₂): for each p(x₁,x₂) there exists p(u|x₁) s.t. |

|≦|

₁ |, R ₁ ≧I(X ₁ ;U), R ₀ ≧I(X ₁ ,X ₂ ;Y|U)}

₂={(R ₀ ,R ₁ ,R ₂): for each p(x₁,x₂) there exists p(u|x₂) s.t. |

|≦|

₂ |, R ₂ ≧I(X ₂ ;U), R ₀ ≧I(X ₁ ,X ₂ Y|U)}.

Then for any (R₀,R₁,R₂)∈

the bit-pipe model shown in FIG. 5c is an inner bounding bit-pipe modelfor

, and for any (R₀,R₁,R₂)∈

the bit-pipe model shown in FIG. 5c is an outer bounding bit-pipe modelfor

.

Consider a memoryless interference channel,

=(

₁×

₂,p(y₁,y₂|x₁,x₂),

₁×

₂). Any existing achievability scheme for that interference channelrepresents an inner bounding model. An achievability scheme that canreliably deliver rate R₁ from transmitter 1 to both receivers, rate R₂from transmitter 1 to receiver 1, rate R₅ from transmitter 1 to receiver2 rate R₇ from transmitter 2 to both receivers, rate R₈ from transmitter2 to receiver 1, and rate R₉ from transmitter 2 to receiver 2 providesan inner bounding model of the form shown in FIG. 5d with R₃=R₄=R₆=0.Let

={(R ₁ , . . . , R ₉): for any p(x₁,x₂), ∃p(u₁,u₂|x₁) s.t. R ₁ ≧I(X ₁ ;U₂), R ₂ ≧I(X ₁ ;U ₁ |U ₂), R ₃ ≧I(X ₁ ,X ₂ ;Y ₂ |U ₂), R ₄ ≧I(X ₁ ,X ₂;Y ₁ |U ₁ ,U ₂ ,Y ₂)}

={(R ₁ , . . . , R ₉): for any p(x₁,x₂), ∃p(u₁,u₂|x₁) s.t., R ₁ ≧I(X ₁ U₁), R ₅ ≧I(X ₁ ;U ₂ |U ₁), R ₃ ≧I(X ₁ ,X ₂ ;Y ₁ |U ₁), R ₆ ≧I(X ₁ ,X ₂;Y ₂ |U ₁ ,U ₂ ,Y ₁)}

={(R ₁ , . . . , R ₉): for any p(x₁,x₂), ∃p(u₁,u₂ 651 x₂) s.t. R ₇ ≧I(X₂ ;U ₂), R ₈ ≧I(X ₂ ;U ₁ |U ₂), R ₃ ≧I(X ₁ ,X ₂ ;Y ₂ |U ₂), R ₄ ≧I(X ₁,X ₂ ;Y ₁ |U ₁ ,U ₂ ,Y ₂)}

={(R ₁ , . . . , R ₉): for any p(x₁,x₂), ∃p(u₁,u₂|x₂) s.t. R ₇ ≧I(X ₂ ;U₁), R ₉ ≧I(X ₂ ;U ₂ |U ₁), R ₃ 624 I(X ₁ ,X ₂ ;Y ₁ |U ₁), R ₆ ≧I(X ₁ ,X₂ ;Y ₂ |U ₁ ,U ₂ ,Y ₁)}

Then for any (R₁, . . . , R₉)∈

, the model shown in FIG. 5d is an outer bounding model.

Equivalent and bounding models for the capacity problem are oftenderived through the solution of information theoretic problems such asthose described in the above examples. In some cases, solutions to thoseinformation theoretic problems are solved analytically and placed into adictionary of component models. Some embodiments of the invention employcomputational tools to derive new component models and add thosecomponent models to the library when the library contains no priorcomponent model for a component of interest. For example, given achannel

=(

,p(y|x),

) for which there is no bounding model currently in the dictionary, someembodiments employ the Blahut-Arimoto algorithm to derive the channelcapacity, which implies an equivalent model as described above.Solutions for other information theoretic characterizations, such as thecharacterizations corresponding to the broadcast, multiple access, andinterference channels can likewise be derived computationally using themethodologies described in W.-H. Gu and M. Effros, “On approximating therate region for source coding with coded side information,” In theProceedings of the IEEE Information Theory Workshop, pages 432-435, LakeTahoe, Calif., September 2007 and W.-H. Gu and S. Jana and M. Effros,“On approximating the rate regions for lossy source coding with codedand uncoded side information”, In the Proceedings of the IEEEInternational Symposium on Information Theory, pages 2162-2166, Toronto,Canada, July 2008. The disclosure of the papers by W.-H. Gu and Effrosand W.-H. Gu et al. is incorporated by reference herein in its entirety.

Jalali and Effros have established similar results for the jointsource-channel coding problem to those described above for the capacityproblem. They show first that an arbitrary collection of jointsource-channel connections can be met on a network of noisy,independent, memoryless links if and only if it can be met on anothernetwork where each noisy link is replaced by a noiseless bit pipe withcapacity equal to the noisy link capacity. The interchangeability ofnoisy, memoryless links with noiseless bit pipes for this problem isdescribed in the papers: S. Jalali and M. Effros, “On the separation oflossy source-network coding and channel coding in wireline networks,”Proceedings of the IEEE International Symposium on Information Theory,pages 500-504, Austin, Tex., June 2010; S. Jalali and M. Effros, “On theseparation of lossy source-network coding and channel coding in wirelinenetworks” ArXiv e-prints, May 2010; and S. Jalali and M. Effros,“Separation of source-network coding and channel coding in wirelinenetworks,” Information Theory and Applications Workshop, San Diego,Calif., February 2011. They then demonstrate general conditions underwhich a bit-pipe model can be shown to be an inner or outer boundingmodel from the perspective of joint source-channel connections. Theseresults appear in S. Jalali and M. Effros, “On distortion bounds fordependent sources over wireless networks” IEEE International Symposiumon Information Theory, Saint Petersburg, Russia, August 2011. Thedisclosure of the papers by Jalali and Effros referenced above areincorporated by reference herein in their entirety.

Although specific components and equivalent and bounding bit-pipe modelsfor specific metrics are discussed in the above referenced papers,network analyzers can utilize bounding and equivalent bit-pipe models toreplace additional and/or alternative components of stochastic networkmodels for the same and other metrics as appropriate to specificapplications in accordance with embodiments of the invention.

Component Modification: Bit-pipe Components

In several embodiments where part or all of either the original networkor an equivalent or bounding network obtained in a prior step is anoiseless, acyclic, wireline network, a network analyzer can applycomponent modifications aimed at simplifying the graphical model. Insome instances, the modifications result in fewer links and/or nodes inthe component model. In other instances, the modifications result inmore links and/or nodes but aid network evaluation or establish anopportunity for greater network simplification in a later iteration ofthe modification process. In both cases, the replacement structure hasthe same inputs and outputs as the original . In this way, boundingmodels for the component preserve bounds for all connections whoseinputs and outputs are outside the modified component.

Consider two networks of noiseless bit pipes,

=(

,

) and

′=(

′,

). Network

may be the original network, a bounding network for the originalnetwork, or a subnetwork of either an original network or a boundingnetwork. For each edge e∈

, the capacity of bit pipe e is C_(e). For each edge e′∈

, the capacity of bit pipe e′ is C_(e′). Network

has a set I⊂

of input nodes which have no incoming links in

, and a set O⊂

of output nodes which have no outgoing links in

. Each node v∈I∪O maps to a node φ(v)∈

′. Two nodes v, v′∈I∪O may, potentially, map to the same node

′, in which case φ(v)=(v′). One way to prove that network

′ is an outer bounding model

for is to demonstrate that every function f=(f_(t):t∈O) of the inputsincoming to the source nodes v∈I that can be reconstructed at eachrespective sink node t∈O can also be reconstructed at the correspondingsinks (φ(t):t∈O) when the same inputs are applied to the nodes(φ(v):v∈I) in

′. One way to show that network

′ is an inner bounding model for

is to show that

is an outer bounding model for

′. The given definitions apply to a variety of network metrics includingthe capacity and joint source-channel metrics. If

′is both an outer bounding model and an inner bounding model for

, then

and

′ are equivalent.

One possible operation for obtaining inner or outer bounding networks isreducing or increasing individual link capacities. As a special case ofsuch operations, one can reduce the capacity of a link to zero, which isthe same as deleting the link. In some cases reducing/increasing linkcapacities is helpful in simplifying the network. For example, onesimplification operation that can be performed in accordance withembodiments of the invention involves removing a set

of links or components to obtain an inner bound. An outer bound with thesame topology can be obtained by adding sufficient capacity to theremaining network to be able to carry any information that could havebeen carried by the set

.

In some instances of the invention, an inner or outer bounding networkis found with the aid of an optimization algorithm. For example, in someembodiments, a linear program is applied to find the minimum factor k bywhich uniformly scaling up the capacities of one network gives an outerbounding model of another network. For example, consider two networks ofnoiseless bit pipes,

=(

,

) and

′=(

,

′) as described above. Associate a commodity with each link e∈

. A path segment including multiple adjacent links without intermediatebranching points is treated as a single logical link with a singleassociated commodity; the effective capacity of that composite linkequals the capacity of the minimum of the constituent links' capacities.The ratios of the capacities of incoming and outgoing links of a node v∈

define the ratios with which the commodities associated with incominglinks are coded together to form commodities associated with theoutgoing links. For each pair (e,e′)∈

×

′, let f_(e,e′) be the fraction of the flow on link e∈

that is carried on link e′∈

′. For each pair (v,v′)∈

×≲′, let r_(v,v′) be the fraction of the coding operations associatedwith node v∈

that are carried out at node v′∈

′. The following linear program finds the minimum factor k by whichscaling up the capacities of links in

′ uniformly gives an outer bounding network for

.

Minimize k subject to

  r_(v, φ(v)) = 1∀v ∈ I⋃O   0 ≤ f_(e, e^(′)) ≤ 1∀(e, e^(′)) ∈ ×  0 ≤ r_(v, v^(′)) ≤ 1∀(v, v^(′)) ∈  × ^(′)  f_(e, e^(′))C_(e) ≤ k C_(e^(′))∀e^(′)∈${{\sum\limits_{e^{\prime} \in {{In}{(v^{\prime})}}}f_{e,e^{\prime}}} + r_{{{Start}{(e)}},v^{\prime}}} = {{\sum\limits_{e^{\prime} \in {{Out}{(v^{\prime})}}}f_{e,e^{\prime}}} + {r_{{{End}{(e)}},v^{\prime}}{\forall{\left( {e,v^{\prime}} \right) \in {\times {^{\prime}.}}}}}}$

If the network

can be obtained from network

by removing some collection of edges, then

′ is an inner bounding model for

, and the network

″ with vertices

′, edges

′ and edge capacities kC_(e′) for each edge e′∈

′ is an outer bounding model for

. Here k is the output of the above linear program. This gives amultiplicative bound of k on the potential capacity differenceassociated with the inner and outer bounding operations, as discussed ina later section.

As is clear to one of ordinary skill in the art, the above algorithmgeneralizes to optimize the individual link capacities for an arbitraryouter bounding model topology. In particular, replacing kC_(e′) (here kis a variable and C_(e′) is fixed) in the above equations by a variableC_(e′) (to be optimized by the algorithm) creates a new set of equationsthat is linear in the variables C_(e′), e′∈

′. Applying an optimization algorithm to optimize an objective functionthat relies on these variables subject to the given constraints providesa mechanism for optimizing outer bounding models subject to a variety ofobjective functions. For example, applying a linear programmingalgorithm to optimize a Lagrangian sum of the edge capacities C_(e′)gives an outer bounding model on the outer convex hull of the space ofsuch models.

Another type of operation for simplifying networks are operations basedon network cut-sets. A cut P=(

₁,

₂) between two sets of nodes

₁ and

₂ is a partition of the network nodes

into two sets

₁ and

₂ such that

₁ ⊂

₂. The capacity of a cut is defined as the sum of capacities of theforward links of the cut, i.e., links (v,w) such that v∈

₁, w∈

₂, giving

C_(P) = C_((v, w)).

A minimal cut P* between sets

₁ and

₂ is a cut P between

₁ and

₂ that minimizes capacity. That is,

$P^{*} = {\arg \; {\min\limits_{{{({_{1},_{2}})}:{_{1} \subseteq _{1}}},{_{2} \subseteq _{2}},{{_{1} \subseteq _{2}} = \varnothing},{{_{1}\bigcup _{2}} = }}{C_{({_{1},_{2}})}.}}}$

Links (v,w)∈

such that v∈

₂ and w∈

₁ are called backward links of cut (

₁,

₂). One example of an operation based on network cuts is as follows. Ifa minimum cut can be found separating a sink from its correspondingsources and all other sink nodes, and the forward links can be directlyconnected to the sink node while preserving all of the backward links,then an outer-bounding network results. If instead of keeping thebackward links, the backward links are deleted, an inner-boundingnetwork is obtained. In the case where there are no backward links, thisprocedure results in an equivalent network. In many embodiments, thisprocedure is repealed for every sink.

The simplification operations described above are described in detail inT. Ho, M. Effros, and S. Jalali, “On equivalence between networktopologies”, Allerton Annual Conference on Communications, Control, andComputing, Monticello, Ill., September 2010 and M. Effros, T. Ho, and S.Jalali, “On Equivalence Between Network Topologies”, ArXiv e-prints,October 2010, the disclosure of which is incorporated herein byreference in its entirety. The processes described in those papersgeneralize to large classes of network topologies. For example, considerthe network

₁ (160) shown in FIG. 6. The Y-network Y(Σa_(i),Σb_(i),Σc_(i))(170)shown in FIG. 7 is an outer bound for the network

. If the network analyzer can determine that

contains two capacity-disjoint Y-networks of the formY(αΣa_(i),αΣb_(i),αΣc_(i)) and Y((1−α)Σa_(i),(1−α)Σb_(i),(1−α)Σc_(i)),then Y(Σa_(i),Σb_(i),Σc_(i)) is also an inner bound for, and thusequivalent to,

. This construction can also be generalized by replacing the basicT-shaped topology by other topologies with arbitrary numbers of inputsand outputs. The star-shaped topology is one simple example.

Although specific simplification operations are described above, any ofa variety of simplification operations can be utilized that enable thereplacement of a subnetwork within a bit-pipe model with bounding modelsthat simplify the calculation of the capacity bounds of the network inaccordance with embodiments of the invention.

In some instances, replacements are performed hierarchically, replacingportions of the network that resulted from a prior analysis with a modelderived in a later analysis. Bounding the accuracy of inner and outerbounding networks is easier when the topologies of the inner and outerbounding networks match. As a result, in some instances of the currentinvention, the analysis performed by the network analyzer applies innerand outer bounding models with the same topologies.

Although the analysis described above utilizes an acyclic graph,analysis can also be applied in graphs with cycles. In some instances ofthe current invention, the cyclic network is replaced by an acyclictime-expanded network and the analysis is applied to that network.

Besides its generality, another benefit of the proposed recursivegraph-based approach is that it allows computation to be traded offagainst tightness of obtained bounds in accordance with the requirementsof a specific application. In some embodiments, analysis is applieddirectly to the noisy network either before or after noisy channelmodeling or sub-network modeling. For example, replacing two nodes in anetwork with a single node that receives all channel outputs andcontrols all channel inputs from the removed nodes always yields anouter bounding model. Likewise, removing one or more channel outputs atany node in a network always yields an inner bounding model. In someembodiments, such simplifications are applied before component modeling.Each of these techniques applies for both bit-pipe models andprobabilistic models and each is employed in some embodiments. Althoughspecific processes are described above for modification of bit-pipecomponents, any of a variety processes appropriate to specificapplications can be utilized to modify bit-pipe components in accordancewith embodiments of the invention.

Bounding Network Performance

Referring back to FIG. 1, the network model output by a componentmodification process (106), similar to one of the component modificationprocesses described above, can be characterized (108) in order tocharacterize the performance of the original network. In manyembodiments, the characterization (108) involves bounding theperformance of the network model obtained using the componentmodification process. Processes for bounding network performance inaccordance with embodiments of the invention are discussed below.

Network analyzers in accordance with many embodiments of the inventioncalculate or bound the performance of a bounding or equivalent network.In some cases, the performance of a network of error-free capacitatedlinks can be obtained analytically. In several embodiments, theperformance of the bounding or equivalent network is calculatedanalytically—thereby calculating or bounding the performance of theoriginal network. (Analytical calculation of the performance of an inner(outer) bounding network guarantees an inner (outer) bound on theperformance of the original network.) In several embodiments, thecapacity of the bounding networks is computed analytically usingtechniques similar to those described in M. Effros, “Capacity bounds fornetworks of broadcast channels” Proceedings of the IEEE InternationalSymposium on Information Theory, pages 580-584, Austin, Tex., June 2010,the disclosure of which is incorporated by reference herein in itsentirety.

In several embodiments, when the connection types fall in the family ofconnection types for which cut-set bounds on capacity are tight,capacity of a bounding network is computed by calculating capacitycut-set bounds on the equivalent/bounding network.

In many embodiments, the network analyzer bounds the performance of theoriginal network by bounding the performance of the bounding network.Inner (outer) bounds on the performance of inner (outer) boundingnetworks guarantee inner (outer) bounds on the original network. Any ofa number of computational techniques can be utilized to give a boundingcharacterization of a network. For example, when the bounding model is abit-pipe network and the metric of interest is capacity, possibletechniques include (but are not limited to) those disclosed in thefollowing papers: L. Song, R. W. Yeung, and N. Cai, “Zero-error networkcoding for acyclic networks.” IEEE Transactions on Information Theory,52(6):2345-3139, July 2003; N. Harvey, R. Kleinberg, and A. R. Lehman,“On the capacity of information networks.” IEEE Transactions onInformation Theory, 52(6):2345-2364, June 2006; and A. T. Subramanianand A. Thangaraj, “A simple algebraic formulation for the scalar linearnetwork coding problem” ArXiv e-prints, July 2008. The disclosure of thepapers by Song et al., Harvey et al, and Subramanian and Thangaraj arehereby incorporated by reference in their entirety. The specifictechnique selected for bounding the network of error-free capacitatedlinks depends upon the specific network and the requirements of thespecific application.

Inner bounds on network performance can be obtained by restrictingattention to special families of codes and then finding the performanceof any code within the given family. The tightest inner bound for agiven family of codes is achieved by the best code in the given family.Examples of coding families include routing solutions, scalar linearcodes, vector linear codes, abelian codes, and matroidal codes.Techniques for finding inner bounds include but are not limited to thetechniques described in the following papers, which are incorporated byreference herein in their entirety. The paper Cannons, Dougherty, andZeger, “Network routing capacity,” IEEE Transactions on InformationTheory, 52(3):777-78S, March 2006 includes one example of an innerbounding technique for network capacity on networks of lossless links.The Cannons et al. paper is incorporated by reference herein in itsentirety. The routing capacity describes the rates at which informationcan be delivered through a network using the best routing solution.Since the routing solution is achievable, the routing capacityguarantees an inner bound. The routing capacity can be found using alinear programming solution, as described in the referenced work. Thepaper T. Ho, “Polynomial-time algorithms for coding across multipleunicasts,” ArXiv e-prints, April 2006 (the disclosure of which isincorporated herein by reference in its entirety) describes apolynomial-time algorithm for finding an optimal code among codes thatemploy only pairwise XOR coding operations. The algorithm uses a linearprogramming approach to find a code in the given class. Given acollection of unicast connections (that is, a collection of connections

such that in each connection a single sender sends information to asingle receiver (|T|=1 for all s=(v,

)∈

)) and a corresponding vector of rates, the algorithm succeeds infinding a solution for those connections at those rates if there existsa solution of a slightly higher rate. As a result, the algorithm can beused to find inner bounds on the multiple unicast capacity region. Whilethe algorithm, as described in that work, focuses on finding solutionsto multiple unicast connections, it can also be applied to find innerbounds for general connections using a result from the paper R.Dougherty and K. Zeger “Nonreversibility and equivalent constructions ofmultiple-unicast networks,” IEEE Transactions on Information Theory,52(11):5067-5077, November 2006 (the disclosure of which is herebyincorporated by reference in its entirety). That paper shows that forany network with arbitrary connection set

(e.g.,

may contain both unicast connections ((x₁,

₁)∈

s.t. |

₁|=1) and multicast connections (x₂,

₂)|

s.t. |

₂|>1)), there exists another network

′ and connection set

′ with only unicast connections (

′={(s₁,

₁), . . . , (s_(k),

_(k))} s.t. |

_(i)|=1∀1≦i≦k) such that connections

can be established on network

if and only if connections

′ can be established on network

′. The argument is constructive—meaning that given any network

and demands

, the paper by Dougherty and Zeger describes an explicit technique forderiving network

′ and its corresponding multiple unicast demands

. Running the algorithm by Ho to find an inner bound on the multipleunicast capacity

(

′,

′) of the resulting network then gives an explicit inner bound on thecapacity under general demands for the original network. The paper bySubramanian and Thangaraj referenced above describes an algebraicformulation for characterizing scalar linear network coding solutions.Here each solution corresponds to a polynomial of degree less than orequal to two; solving the polynomial equation for a desired collectionof rates demonstrates the achievability of those rates using scalarlinear codes. Again, the complexity of the solution grows with thenumber of edges in the network.

In a number of embodiments, the network analyzer employs an optimizationalgorithm to find an inner or outer bound on the joint source-channelcoding region. In some embodiments, the core of the algorithm is alinear program. Two examples of such linear programs are described inthe paper M. Effros, “On source and networks: can computational toolsderive information theoretic limits” Allerton Conference onCommunication, Control, and Computing, 2011. Let

=(

,

) be a directed, acyclic graph in which each e∈

is a noiseless bit pipe of capacity C_(e). Let

be a collection of connections. Let p(u) be a distribution on sourcevector (U_(s):s∈

). The network analyzer can find an outer bound on the collection ofdistortion vectors (D_(s,t):s=(v,T)∈

,j∈T), where distortion D_(s,t) is an expected distortion with respectto distortion measure d_(s,t)(U_(s):s∈

) that can be achieved across network

under source distribution p(u) of network

. Let (U_(s):s∈

) be the source vector, (U_(e):e∈

) be the vector of edge messages, and (Û_(s,t):s=(v,

)∈

,t∈

) be a collection of reconstructions at the receivers. Then for each e∈

, U_(e) is the message traversing edge e, and for each s=(v,

)∈

and each t∈

, Û_(s,t) is the reconstruction of source U_(s) at receiver t. For eachsubset A⊂

, U_(A)=(U_(s):s∈A) and H(U_(A)) is the entropy of U_(A) underdistribution p. The alphabet of U_(s) for each s∈

may be discrete or continuous. For each v∈

, let

U _(v)=((U _(s) :s=(v,

)∈

), (U _(e) :e∈In(v)));

that is, U_(v) describes all inputs (both sources and messagestraversing incoming edges) to node v. In the linear program, there is avariable h_(A) for each subset A of the set

={U_(s):s∈

}∪{U_(e): e∈

}∪{Û_(s,t):s=(v,

)∈

,t∈

}. For notational simplicity, the set brackets for set A are oftendropped; thus h_({U) _(s) _(,U) _(v) _(}) and h_(U) _(s) _(,U) _(v)refer to the same variable. Let D_(s,t)(R) be the distortion-ratefunction for source s according to distortion measure d_(s,t)(·,·). Thespace of distortion vectors (D_(s,t)(h_(U) _(s) +h_(Û) _(s,t) −h_(U)_(s) _(,Û) _(s,t) ):s=(v,

)∈

,t∈

) for which

h _(A) =H(U _(A))∀A ⊂

, |A|>0

h _(U) _((v,v′)) _(,U) _(v) −h _(U) _(v) =0∀(v,v′)∈

h _(Û) _(s,t) _(,U) _(t) −h _(U) _(t) =0∀s=(v,

)∈

,t∈

h _(U) _(e) ≦C _(e) ∀e∈

h _(A)≧0∀A ⊂

\

, |A|>0

h _(A) −h _(B)≧0∀B⊂A ⊂

, |B|>0

h _(A∪C) +h _(B∪C) −h _(A∪B∪C) −h(C)≧0∀A,B,C ⊂

, |A|,|B|,|C|>0

provide an outer bound on the joint source-channel coding region

(

, p,

). In many embodiments, the network analyzer uses linear programming tofind the outer boundary of this region.

In other embodiments, a network analyzer finds an inner boundary on theset of distortions achievable by joint-source-channel coding by findingan inner bound on the capacity region for independent sources underconnections

and evaluating the distortion-rate function at those rates. For example,if rate vector (R_(s):s∈

) is achievable for connections

, then distortion vector (D_(s,t)(R_(s)):s=(v,

),t∈

) with distortion measure d_(s,t)(u,û)=d_(s)(u,û) for each s=(v,

)∈

and all t∈

gives an inner bound on the joint source-channel coding region fornetwork

with (with independent or dependent) sources (U_(s):s∈

).

Although specific processes are described above for bounding networkperformance, any of a variety of processes appropriate to specificapplications can be utilized to determine bounds for the networkperformance in accordance with embodiments of the invention.

Assessing Accuracy of Network Characterizations

Whether the network analysis produces bounds on network performance or abounding model for the network, for many applications it is useful toobtain not only the network analysis but also an assessment of theaccuracy of that analysis. Many network analyzers in accordance withembodiments of the invention include assessments of bounding accuracy aspart of the network analysis.

In some embodiments, these assessments compare the regions computed asinner and outer bounds on the network performance and return one or moremeasures of their closeness. For example, in some embodiments,

_(I)(

,

)and

_(O)(

,

) are inner and outer bounds, respectively, on the capacity region of anetwork

under connections

(

_(I)(

,

)⊂

(

,

)⊂

_(O)(

,

)) found by the network analyzer and

_(O) ^(o)(

,

) is the outer boundary of the outer bounding region. In some instances,the network analyzer bounds the accuracy of the given bounds with aworst measure of accuracy such as (but not limited to)

${d\left( {{_{I}\left( {,} \right)},{_{O}\left( {,} \right)}} \right)} = {\max\limits_{r_{O} \in {_{O}{({,})}}}{\min\limits_{r_{I} \in {_{I}{({,})}}}{{d\left( {r_{O},r_{I}} \right)}.}}}$

or an average measure of accuracy such as (but not limited to)

${{d\left( {{_{I}\left( {,} \right)},{_{O}\left( {,} \right)}} \right)} = \frac{\int_{r_{O} \in {_{O}^{o}{({,})}}}{\min_{r_{I} \in _{O}}{{\left( {r_{O},r_{I}} \right)}{r_{O}}}}}{\int_{r_{o} \in {_{O}^{o}{({,})}}}{r_{O}}}},$

where d(r,r′) captures some notion of distance or distortion (e.g.,d(r,r′)=∥r−r′∥).

When the network analysis derives or employs inner and outer boundingmodels

_(I) and

_(O) for a network

, network analyzers in accordance with several embodiments of theinvention bound the accuracy of the bounding models by comparing theinner and outer bounding networks to each other, Here network

may be a network, a component of a larger network, or a bounding modelfor either a network or a component. In some instances, the networkanalyzer bounds the difference between the performance of a network orcomponent

and the performance of models

_(I) and

_(O) using a bound such as (but not limited to) the bounds described inthe paper M. Effros, “On capacity outer bounds for a simple family ofwireless networks”, Information Theory and Applications Workshop, SanDiego, Calif., 2010, the disclosure of which is incorporated herein byreference in its entirety. As is discussed further below, the processfor obtaining inner and outer bounding networks for subnetworks within ageneral network depends upon the nature of the network.

When comparing inner and outer bounding networks

_(I) and

_(O) for which the stochastic elements of ⊥_(I) and

_(O), if any, are identical and the topologies of all graphicalcomponents of

_(I) and

_(O) are identical ((

_(I),

_(I))=(

_(O),

_(O))=(

,

)) though the capacities C_(e)(

_(I)),C_(e)(

_(O))>0 for all e∈

may differ, the following difference factor between

_(I) and

_(O) can be used

${\delta \left( {_{I},_{O}} \right)} = {{\frac{C_{e}\left( _{I} \right)}{C_{e}\left( _{O} \right)}.}}$

Note that δ(

_(I),

_(O))∈(0,1], and δ(

_(I),⊥_(O))=1 if and only if the inner and outer bounds are identicaland therefore equivalent to the network

. Given a fixed connection set

, let

denote the capacity region under connections

either of the network

or of a larger network of interest that contains

as a component. Let

_(I)(

_(O)) denote the capacity region with respect to the same connections ofthe network that results when

is replaced by

_(I)(

_(O)) in the network corresponding to

. Then

_(I) ⊂

⊂

_(O),

while

δ(

_(I),

_(O))·

_(O) ⊂

_(I),

For any integer k, any set A⊂IR^(k) of k-dimensional real vectors, andany constant c, the set c′A is here defined as c·A={(cr₁. . . ,cr_(k)):(r₁, . . . , r_(k))∈A}. As a result,

δ(

_(I),

_(O))·

_(O) ⊂

⊂

_(O)

_(I) ⊂

⊂(1/δ(

_(I),

_(O)))·

_(I).

In some embodiments of the network analyzer, δ(

_(I),

_(O)) is used as a measure of bounding accuracy.

For some connection sets

, cut-set bounds on the capacity are tight for networks of noiselesslinks. For other connection sets

, cut-set bounds on the capacity are not tight but provide a simplecharacterization of network properties, estimates of networkperformance. As a result, cut-set bounds are also used in some instancesof the network analyzer as measures of the differences between inner andouter bounding models for a network or network component. Consider anarbitrary network or component

. Let

_(I) and

_(O) be inner and outer bounding models for

, where

_(I) and

_(O) are made entirely of noiseless bit pipes. Let

be the connection set of interest, and define the input set

as

M={v:∃

≠ s.t. (v,

)∈

} and the output set

′ as

′={t:∃(v,

)∈

s.t. t∈

}. Given any non-empty, non-intersecting sets A⊂

and B⊂

′, let P_(I)(A,B) be the set of cut sets between A and B in

_(I) and

_(O)(A,B) be the set of cut sets between A and B in

_(O). Then

${\rho \left( {_{I},_{O}} \right)} = {\max\limits_{{{({A,B})}:{A \subseteq \mathcal{M}}},{B \subseteq \mathcal{M}^{\prime}},{{A\bigcap B} = \varnothing}}\left\lbrack {{\min\limits_{P \in {_{O}{({A,B})}}}{C_{P}\left( _{O} \right)}} - {\min\limits_{P \in {_{I}{({A,B})}}}{C_{P}\left( _{I} \right)}}} \right\rbrack}$

measures the maximal difference in minimal cut capacities for all cutsbetween input and output nodes in

_(I) and

_(O). Note that ρ(

_(I),

_(O))≧0 by definition of inner and outer bounding networks. Let

denote the capacity region either of the network

or of a larger network of interest that contains

as a component. Let

_(I)(

_(O)) denote the capacity region of the network that results when

is replaced by

_(I)(

_(O)) in the network corresponding to

. If capacities

,

_(I) and

_(O) are taken with respect to some connection type for which cut-setbounds are tight, then

_(I) ⊂

⊂

_(O),

while

(

_(O)−ρ(

_(I),

_(O))·1)⁺ ⊂

_(I),

where for any integer k, any set A⊂IR^(k), and any constant c,

(A−c·1)⁺={((r ₁ 631 c)⁺, . . . , (r _(k) −c)⁺):(r ₁ , . . . , r _(k))∈A}

and (a)⁺ equals a if a≧0 and 0 otherwise. As a result,

(

_(O)−ρ(

_(I),

_(O))·1)⁺ ⊂

∈

_(O)

_(I) ⊂

⊂

_(I)+ρ(

_(I),

_(O))·1,

where for any c≧0,

A+c·1={(r ₁ +c . . . , r _(k) +c):(r ₁ , . . . , r _(k))∈A}.

In some embodiments of the network analyzer, ρ(

_(I)

_(O)) is used as a measure of bounding accuracy.

The paper M. Effros, “On capacity outer bounds for a simple family ofwireless networks,” Information Theory and Applications Workshop, SanDiego, Calif., 2010, (incorporated above) describes some of the abovemeasures of accuracy.

In some embodiments of the network analyzer, the analyzer bounds theaccuracy of the model by bounding the impact on the networkcharacterization of one or more noiseless capacitated links. Forexample, the paper S. Jalali, M. Effros, T. Ho, “On the impact of asingle edge on the network coding capacity,” Information Theory andApplications Workshop, San Diego, Calif., 2011, incorporated herein byreference in its entirety, bounds the maximal difference in capacitybetween a network and another network that is identical except for theremoval of a single noiseless, capacitated link.

In many embodiments, network analyzers determine whether the accuracy ofthe network performance bound is below a predetermined threshold andrepeat the process of deriving new bounding models in an attempt toimprove that accuracy. Although specific processes are described abovefor determining the network performance and bounding assessmentaccuracy, any of a variety processes appropriate to specificapplications can be utilized to determine bounds for the networkperformance and to bound assessment accuracy in accordance withembodiments of the invention.

While the above description contains many specific embodiments of theinvention, these should not be construed as limitations on the scope ofthe invention, but rather as an example of one embodiment thereof. Forexample, although many embodiments replace noisy links with boundingbit-pipe models, several embodiments utilize simplification tools thatare directly applicable to noisy networks. Accordingly, the scope of theinvention should be determined not by the embodiments illustrated, butby the appended claims and their equivalents.

What is claimed is:
 1. A network analyzer configured to characterize acommunication network, comprising: a processor configured to obtain aninitial model of the communication network; wherein the processor isfurther configured to perform at least one substitution that replaces acomponent of the initial network model with an equivalent or boundingmodel to produce a simplified network that is capable of beingcharacterized by the network analyzer; and wherein the processor is alsoconfigured to characterize the initial model by characterizing thesimplified network model.